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Theorem icomnfordt 17234
Description: An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
icomnfordt  |-  (  -oo [,) A )  e.  (ordTop `  <_  )

Proof of Theorem icomnfordt
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )
2 eqid 2404 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )
3 eqid 2404 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 17231 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 17224 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2475 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 16990 . . . . . . 7  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  <-> 
( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top )
86, 7mpbir 201 . . . . . 6  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases
9 bastg 16986 . . . . . 6  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  ->  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
) )
108, 9ax-mp 8 . . . . 5  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
1110, 4sseqtr4i 3341 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3470 . . . . 5  |-  ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) ) 
C_  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
13 ssun2 3471 . . . . . 6  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )
14 eqid 2404 . . . . . . . 8  |-  (  -oo [,) A )  =  ( 
-oo [,) A )
15 oveq2 6048 . . . . . . . . . 10  |-  ( x  =  A  ->  (  -oo [,) x )  =  (  -oo [,) A
) )
1615eqeq2d 2415 . . . . . . . . 9  |-  ( x  =  A  ->  (
(  -oo [,) A )  =  (  -oo [,) x )  <->  (  -oo [,) A )  =  ( 
-oo [,) A ) ) )
1716rspcev 3012 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  (  -oo [,) A )  =  (  -oo [,) A
) )  ->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
1814, 17mpan2 653 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
19 eqid 2404 . . . . . . . 8  |-  ( x  e.  RR*  |->  (  -oo [,) x ) )  =  ( x  e.  RR*  |->  (  -oo [,) x ) )
20 ovex 6065 . . . . . . . 8  |-  (  -oo [,) x )  e.  _V
2119, 20elrnmpti 5080 . . . . . . 7  |-  ( ( 
-oo [,) A )  e. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )  <->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
2218, 21sylibr 204 . . . . . 6  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )
2313, 22sseldi 3306 . . . . 5  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) ) )
2412, 23sseldi 3306 . . . 4  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3306 . . 3  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
2625adantl 453 . 2  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
27 df-ico 10878 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
2827ixxf 10882 . . . . 5  |-  [,) :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5555 . . . 4  |-  dom  [,)  =  ( RR*  X.  RR* )
3029ndmov 6190 . . 3  |-  ( -.  (  -oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A
)  =  (/) )
31 0opn 16932 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 8 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2492 . 2  |-  ( -.  (  -oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A
)  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 158 1  |-  (  -oo [,) A )  e.  (ordTop `  <_  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667    u. cun 3278    C_ wss 3280   (/)c0 3588   ~Pcpw 3759    e. cmpt 4226    X. cxp 4835   ran crn 4838   ` cfv 5413  (class class class)co 6040    +oocpnf 9073    -oocmnf 9074   RR*cxr 9075    < clt 9076    <_ cle 9077   (,)cioo 10872   (,]cioc 10873   [,)cico 10874   topGenctg 13620  ordTopcordt 13676   Topctop 16913   TopBasesctb 16917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-topgen 13622  df-ordt 13680  df-ps 14584  df-tsr 14585  df-top 16918  df-bases 16920  df-topon 16921
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